Monstrous moonshine

In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a certain conformal field theory having the Monster group as symmetries. The conjectures made by Conway and Norton were proved by Richard Ewen Borcherds in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and Generalized Kac–Moody algebras.

History

In 1978, John McKay, found that the first few terms in the Fourier expansion of j(τ) (sequence A000521 in OEIS), with τ denoting the half-period ratio, could be expressed in terms of linear combinations of the dimensions of the irreducible representations of M (sequence A001379 in OEIS) with very small natural coefficients:

$j(\tau) = \frac{1}{{q}} + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + \cdots$

where ${q} = e^{2\pi i\tau}$, and

$\begin{align} 1 & = 1 \\ 196884 & = 196883 + 1 \\ 21493760 & = 21296876 + 196883 + 1 \\ 864299970 & = 842609326 + 21296876 + 2\cdot 196883 + 2\cdot 1 \\ 20245856256&=18538750076+2\cdot842609326+21296876+3\cdot196883+3\cdot1\\ &=19360062527+842609326+2\cdot21296876+3\cdot196883+2\cdot1\\ 333202640600 & =293553734298+2\cdot 18538750076+3\cdot 842609326+2\cdot 21296876+5\cdot 196883+5\cdot 1\\ & =293553734298+19360062527+ 18538750076+2\cdot 842609326+3\cdot 21296876+5\cdot 196883+4\cdot 1\\ 4252023300096 & =3879214937598+293553734298+4\cdot 18538750076+6\cdot842609326+2*21296876+7\cdot196883+7\cdot1 \\ & {}\,\,\, \vdots \end{align}$

McKay viewed this as evidence that there is a naturally-occurring infinite-dimensional graded representation of M, whose graded dimension is given by the coefficients of j, and whose lower-weight pieces decompose into irreducible representations as above. After he informed John G. Thompson of this observation, Thompson suggested that because the graded dimension is just the graded trace of the identity element, the graded traces of nontrivial elements g of M on such a representation may be interesting as well.

Conway and Norton computed the lower-order terms of such graded traces, now known as McKay-Thompson series T_g, and found that all of them appeared to be the expansions of Hauptmoduln. In other words, if G_g is the subgroup of SL₂(R) which fixes T_g, then the quotient of the upper half of the complex plane by G_g is a sphere with a finite number of points removed, and furthermore, T_g generates of the field of meromorphic functions on this sphere.

Based on their computations, Conway and Norton produced a list of Hauptmoduln, and conjectured the existence of an infinite dimensional graded representation of M, whose graded traces T_g are the expansions of precisely the functions on their list.

In 1980, A. Oliver L. Atkin, Paul Fong and Stephen D. Smith, showed that such a graded representation exists, using computer calculation to decompose coefficients of j into representations of M up to a bound discovered by Thompson. A graded representation was explicitly constructed by Igor Frenkel, James Lepowsky, and Arne Meurman, giving an effective solution to the McKay-Thompson conjecture. Furthermore, they showed that the vector space they constructed, called the Moonshine Module $V^\natural$, has the additional structure of a vertex operator algebra, whose automorphism group is precisely M.

Borcherds proved the Conway-Norton conjecture for the Moonshine Module in 1992. He won the Fields medal in 1998 in part for his solution of the conjecture.

The Monster module

The Frenkel-Lepowsky-Meurman construction uses two main tools:

1. The construction of a lattice vertex operator algebra V_L for an even lattice L of rank n. In physical terms, this is the chiral algebra for a bosonic string compactified on a torus Rⁿ/L. It can be described roughly as the tensor product of the group ring of L with the oscillator representation in n dimensions (which is itself isomorphic to a polynomial ring in countably infinitely many generators). For the case in question, one sets L to be the Leech lattice, which has rank 24.
2. The orbifold construction. In physical terms, this describes a bosonic string propagating on a quotient orbifold. The construction of Frenkel-Lepowsky-Meurman was the first time orbifolds appeared in conformal field theory. Attached to the -1 involution of the Leech lattice, there is an involution h of V_L, and an irreducible h-twisted V_Lmodule, which inherits an involution lifting h. To get the Moonshine Module, one takes the fixed point subspace of h in the direct sum of V_L and its twisted module.

Frenkel, Lepowsky, and Meurman showed that the automorphism group of the moonshine module, as a vertex operator algebra, is M, and they showed that its graded dimension gives the Fourier expansion of j.

Borcherds' proof

Richard Borcherds' proof of the conjecture of Conway and Norton can be broken into the following major steps:

1. One begins with a vertex operator algebra V, with an action of M by automorphisms, and with graded dimension j. This was provided by the Moonshine Module, also called the monster vertex algebra or monster VOA.
2. A Lie algebra $\mathfrak{m}$, called the monster Lie algebra, is constructed from V using a quantization functor. It is a generalized Kac-Moody Lie algebra with a monster action by automorphisms. Using the Goddard–Thorn "no-ghost" theorem from string theory, the root multiplicities are found to be coefficients of j.
3. One uses the Koike-Norton-Zagier infinite product identity to construct a generalized Kac-Moody Lie algebra by generators and relations. The identity is proved using the fact that Hecke operators applied to j yield polynomials in j.
4. By comparing root multiplcities, one finds that the two Lie algebras are isomorphic, and in particular, the Weyl denominator formula for $\mathfrak{m}$ is precisely the Koike-Norton-Zagier identity.
5. Using Lie algebra homology and Adams operations, a twisted denominator identity is given for each element. These identities are related to the McKay-Thompson series T_g in much the same way that the Koike-Norton-Zagier identity is related to j.
6. The twisted denominator identities imply recursion relations on the coefficients of T_g. These relations are strong enough that one only needs to check that the first seven terms agree with the functions given by Conway and Norton.

Thus, the proof is completed. Borcherds was later quoted as saying "I was over the moon when I proved the moonshine conjecture", and "I sometimes wonder if this is the feeling you get when you take certain drugs. I don't actually know, as I have not tested this theory of mine."

Generalized Moonshine

Conway and Norton suggested in their 1979 paper that perhaps moonshine is not limited to the monster, but that similar phenomena may be found for other groups. In 1980, Larissa Queen found that one can in fact construct the expansions of many Hauptmoduln of level greater than one from simple combinations of dimensions of sporadic groups, such as the Baby monster. For example, the smallest faithful representation of the Baby monster has dimension 4371, while the Hauptmodul for Γ₀(2)⁺ has expansion $q^{-1} + 4372q + \cdots$.

In 1987, Norton combined Queen's results with his own computations to formulate the Generalized Moonshine conjecture. This conjecture asserts that there is a rule that assigns to each element g of the monster, a graded vector space V(g), and to each commuting pair of elements (g,h) a holomorphic function f(g,h,τ) on the upper half plane, such that:

1. Each V(g) is a graded projective representation of the centralizer of g in M.
2. Each f(g,h,τ) is either a constant function, or a Hauptmodul.
3. Each f(g,h,τ) is invariant under simultaneous conjugation of g and h in M.
4. For each (g,h), there is a lift of h to a linear transformation on V(g), such that the expansion of f(g,h,τ) is given by the graded trace.
5. For any $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z})$, $f(g,h,\frac{a\tau+b}{c\tau+d})$ is proportional to $f(g^a h^c, g^b h^d, \tau)$.
6. f(g,h,τ) is proportional to j if and only if g = h = 1.

This is a generalization of the Conway-Norton conjecture, because Borcherds's theorem concerns the case where g is set to the identity. To date, this conjecture is still open.

Like the Conway-Norton conjecture, Generalized Moonshine also has an interpretation in physics, proposed by Dixon-Ginsparg-Harvey in 1988. They interpreted the vector spaces V(g) as twisted sectors of a conformal field theory with monster symmetry, and interpreted the functions f(g,h,τ) as genus one partition functions, where one forms a torus by gluing along twisted boundary conditions. In mathematical language, the twisted sectors are irreducible twisted modules, and the partition functions are assigned to elliptic curves with principal monster bundles, whose isomorphism type is described by monodromy along a basis of 1-cycles, i.e., a pair of commuting elements.

Mathieu Moonshine

In 2010, Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa observed that conformal field theory on a K3 surface can be decomposed into representations of the N=(4,4) superconformal algebra, such that the multiplicities of massive states appear to be simple combinations of irreducible representations of the Mathieu group M24. By the Mukai-Kondo classification, there is no faithful action of this group on any K3 surface by symplectic automorphisms, and also no action on the corresponding conformal field theories, so the appearance of an action on the underlying Hilbert space is still a mystery.

Additional calculations suggest that both the multiplicity functions and the graded traces of nontrivial elements of M24 form Mock modular forms. In 2012, Gannon proved that all but the first of the multiplicities are non-negative integral combinations of representations of M24, and Gaberdiel et al computed all analogues of generalized moonshine functions. Also in 2012, Cheng, Duncan, and Harvey amassed numerical evidence of an umbral moonshine phenomenon, of which Mathieu Moonshine is a special case, and so far, it does not have an interpretation in terms of geometry such as conformal field theory on K3 surfaces.

Why "monstrous moonshine"?

The term "monstrous moonshine" was coined by Conway, who, when told by John McKay in the late 1970s that the coefficient of ${q}$ (namely 196884) was precisely the dimension of the Griess algebra (and thus exactly one more than the degree of the smallest faithful complex representation of the Monster group), replied that this was "moonshine" (in the sense of being a crazy or foolish idea). Thus, the term not only refers to the Monster group M; it also refers to the perceived craziness of the intricate relationship between M and the theory of modular functions.

However, "moonshine" is also a slang word for illegally distilled whiskey, and in fact the name may be explained in this light as well. The Monster group was investigated in the 1970s by mathematicians Jean-Pierre Serre, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL₂(R), particularly, the normalizer Γ₀(p)⁺ of Γ₀(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ₀(p)⁺ has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg heard about the Monster group later on, and noticed that these were precisely the prime factors of the size of M, he published a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact.

References

• John Horton Conway and Simon P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11, 308–339, 1979.
• I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, 1988
• Richard Ewen Borcherds, Monstrous Moonshine and Monstrous Lie Superalgebras, Invent. Math. 109, 405–444, 1992, online
• Terry Gannon, Monstrous Moonshine: The first twenty-five years, 2004, online
• Terry Gannon, Monstrous Moonshine and the Classification of Conformal Field Theories, reprinted in Conformal Field Theory, New Non-Perturbative Methods in String and Field Theory, (2000) Yavuz Nutku, Cihan Saclioglu, Teoman Turgut, eds. Perseus Publishing, Cambridge Mass. ISBN 0-7382-0204-5 (Provides introductory reviews to applications in physics).
• Gannon, Terry (2006), Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, ISBN 0-521-83531-3 
• Koichiro Harada, Monster, Iwanami Pub. (1999) ISBN 4-00-006055-4, (The first book about the Monster Group written in Japanese).
• Mark Ronan, Symmetry and the Monster, Oxford University Press, 2006. ISBN 978-0-19-280723-6 (Concise introduction for the lay reader).
• Marcus Du Sautoy, Finding Moonshine, A Mathematician's Journey Through Symmetry. Fourth Estate, 2008 ISBN 0-00-721461-8, ISBN 978-0-00-721461-7

External links

• Moonshine Bibliography at the Wayback Machine